Yielding Rod to Counter Seismic Activity

ABSTRACT

A yielding rod to counteract seismic activity having a generally cylindrical shape with a narrow portion having a reduced cross-sectional diameter relative to the remaining portions of the yielding rod. The narrow portion spans a length that is approximately 8 times its diameter to approximately 16 times its diameter. A transition area connects the narrow portion of the rod to the remainder of the rod. The transition area spans a length of approximately 0.25 inch to approximately 1.5 inch and has a preferred slope of 30 degrees or less relative to a longitudinal axis of the rod.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application claims the benefit of U.S. Provisional PatentApplication No. 61/461,495 entitled “Threaded Rod with a ReducedCross-Section,” filed Jan. 19, 2011, which application is incorporatedin its entirety here by this reference.

TECHNICAL FIELD

This invention relates to anchoring rods that counter seismic activity.

BACKGROUND

In the structural design of building structures located in areas of highseismic risk, it is desirable to control the behavior of the structureand thus limit the damage. One method of controlling behavior is bydesigning certain structural components to fail in a controlled anddesirable manner

Building codes are now requiring that structures that are expected toexperience seismic events, behave in a ductile manner That is, astructure must be able to undergo large inelastic deformations withoutlosing its strength. One particularly problematic issue is creating aductile connection at the base of a structure where it is anchored toits concrete foundation. Current building codes require that the failureof anchorage connections be governed by the “steel strength of a ductilesteel element” (ACI 318-08, section D.3.3.4), as opposed to the brittlebehavior associated with failure of concrete or epoxies. This designgoal is currently extremely difficult and expensive to achieve due tothe fact that steel is extremely strong in tension relative to thestrength of concrete and epoxy. It is especially expensive, both interms of costs associated with labor and materials, to achieve this goalin existing structures. This is due to the fact that the geometry of theexisting foundation is already set and cannot be changed with relativeease. It is important to solve this problem, in order to build moredependable, better performing and economical structures.

Structures are generally anchored to a concrete foundation with steelanchor bolts that prevent the structure from lifting away from thefoundation or moving laterally. The types of bolts that are mostcommonly used in new construction are J-bolt, L-bolt, hex head bolt andthreaded rods. For anchor bolts placed in existing hardened concrete,such as would occur in a remodel or retrofit of the building, a commonlyused method of anchoring of building is to drill a hole in the concretefoundation, fill the hole with an adhesive (commonly referred to asepoxy), and then insert a threaded rod into the hole. For buildingslocated in areas of seismic risk current building codes require thatfailure of anchor bolt connections be limited to the failure of aductile steel element and not the concrete or epoxy. While thisrequirement is relatively easy to fulfill in cases of new constructionand were anchor bolts are placed far away from the edges of thesupporting concrete, the same is not true for anchor bolt installationsnear the edges of supporting concrete elements.

The most common solution for new construction involves increasing theamount of concrete and reinforcement around the anchor bolts. Forexisting construction, the most common solution to ensure ductilefailure involves drilling bolts through the foundations, which may beseveral feet deep, and digging such that a nut and washer can be placedat the end of the anchor bolt. The problem with these solutions is thatthey require an increased amount of labor, material, and time to build.

Other fields also utilize anchors designed to withstand or compensatefor lateral impact. However, these anchors are insufficient to use foranchoring a building to a foundation to withstand seismic activity. Thisis because these anchors have reduced cross-sectional areas that aredesigned to snap off upon impact rather than yielding.

For the foregoing reasons there is a need for a method and a device thatwill withstand seismic activity while being economical and efficient toinstall and replace.

SUMMARY

The yielding rod is an anchor device to counteract seismic activityencountered by a building, that also reduces the amount of labor,material, and time required for concrete anchorage connections of thebuilding to its foundation. The yielding rod comprises a cylindrical barhaving a specifically designed reduced cross-sectional area, such thatit counteracts seismic activity by yielding itself rather than throughfailure of the concrete or epoxy. Due to the reduced cross-sectionalarea, seismic activity is less likely to damage the foundation ofbuilding because the reduced cross-sectional area dissipates energythrough yielding in tension during a large earthquake. Even if theearthquake was so powerful as to cause damage, the damage would be tothe reduced cross-sectional area of the yielding rod as opposed to theconcrete foundation. However, replacing or repairing the yielding rodwill result in a tremendous amount of savings in labor, material, andtime because of the ease with which these anchors can be easily replacedor repaired without modifying or reinforcing the concrete foundation.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows an elevation view of an embodiment of the presentinvention;

FIG. 2 shows an elevation view of an embodiment of the present inventionin use; and

FIG. 3 shows an elevation view of an embodiment of the present inventionin use to repair a damaged anchor; and

FIGS. 4A-4C show the stress on a yielding rod at different stages oftension applied to the yielding rod.

DETAILED DESCRIPTION OF THE INVENTION

The detailed description set forth below in connection with the appendeddrawings is intended as a description of presently-preferred embodimentsof the invention and is not intended to represent the only forms inwhich the present invention may be constructed or utilized. Thedescription sets forth the functions and the sequence of steps forconstructing and operating the invention in connection with theillustrated embodiments. It is to be understood, however, that the sameor equivalent functions and sequences may be accomplished by differentembodiments that are also intended to be encompassed within the spiritand scope of the invention.

The present invention is directed towards a yielding rod 100 foranchoring a building or house to a foundation 20, wherein the rod 100 isconfigured to yield under stress, such as that created by an earthquake.The rod 100 is generally cylindrical in shape having a first portion 102terminating at a first end, a second portion 104 terminating at a secondend opposite the first end, the rod 100 defining a longitudinal axis Athat goes through the first and second portions 102, 104. Portions ofthe rod 100 in between the first and second ends taper radially inwardto a narrow portion 106 having a reduced cross-sectional diameter D₁relative to the cross-sectional diameter D₂ of the first and second ends102, 104. Thus, the tapering portions define a first transition area 108connecting the first portion 102 to the narrow portion 106 and a secondtransition area 110 connecting the second portion 104 to the narrowportion 106.

The ductility of the yielding rod 100 is realized by the preciserelationship of the dimensions of the diameter D₁ of the narrow portion106 relative to the length L₁ of the narrow portion 106, and the lengthof the transition areas L₂, L₃. In other applications using rods with areduced cross-sectional diameter, the length of the cross-sectionaldiameter area is much smaller than the diameter of the reducedcross-sectional area, which allows the anchor to snap or break at thereduced cross-sectional diameter area with the application of force.However, the ductility of steel increases with an increase in itslength. Therefore, elongating the length L₁ of a narrow portion 106 andthe lengths L₂, L₃ of the transition areas 108, 110 could improve itsductility and reduce its propensity to snap under lateral forces.

Simply elongating the length L₁ of a narrow portion 106 would notnecessarily result in an effective yielding rod. If the dimension of thediameter D₁ of the narrow portion 106 is not proper, then the yieldingrod 100 may not properly perform its function.

For example, the narrow portion 106 of the yielding rod 100 must have astatic strength in tension that is less than the static breakoutstrength in tension and the static pullout strength in tension of aconcrete foundation 20 in which the yielding rod 100 has been embedded.This will assure that the weakest point of the foundation is the narrowportion 106 of the yielding rod 100 and not the concrete 20. Therefore,during an earthquake, the lateral forces imparted on the foundation bythe building 10 will cause the yielding rod 100 to absorb the tensionrather than causing the yielding rod 100 to break out or pull out fromthe concrete foundation 20, thereby avoiding destruction of the concrete20. With the concrete 20 still intact then, only the yielding rod 100needs to be replaced or repaired.

The static concrete breakout strength in tension and the static pulloutstrength in tension of the concrete foundation can be determined basedon calculations established by the International Code Council and theAmerican Concrete Institute. The lesser of the two values will bedesignated as the minimum concrete strength (MC). The diameter D₁ of thenarrow portion 106 of the yielding rod 100 is then calculated by thefollowing equation:

D ₁=Square root of [(MC)/((pi/4)×(T))],   Eq. 1:

where MC is the minimum concrete strength and T is the upper limit ofthe tensile strength of the yielding rod 100 material (e.g. steel).

With the requisite diameter D₁ of the narrow portion 106 known, thedimensional ranges for the transition lengths L₂, L₃ can be determinedby performing a non-linear finite element analysis of a wide range ofgeometries. The transition lengths L₂, L₃ are the distance along thelongitudinal axis A covered by a transition area 108, 110. Thetransition lengths L₂, L₃ must be long enough such that it helps removeany concentrated stresses, but short enough so that it does not take uptoo much space. In developing the models, several long slender rods ofsteel were tested in order to determine the mechanical properties of thesteel. In a first analysis, a steel rod with transition lengths L₂, L₃of 1.5 inch was used. The mechanical properties were plotted in a graphdepicting the stress versus strain with the stress representing theforce imparted on the rod along its longitudinal axis and the strainrepresenting the length of elongation of the narrow portion 106. Thedata points from the stress-strain curve were then input into thecomputer model in order to realistically depict the behavior of theinvention.

A second analysis was then run with each transition length L₂, L₃ of 0.5inch and a length L₁ for the narrowed portion 106 at 8 times itsdiameter D₁, which established the upper bound of angles for thetransition areas 108, 110.

A third analysis was run with a 0.5 inch transition lengths, L₂, L₃ andvariable lengths L₁ for the narrowed portion 106 to study the effects oflengthening the narrowed portion 106.

Tables 1-3 shows some results of the experiment correlating the yieldingcharacteristics of a steel yielding rod 100 to different parameters ofthe length L₁ of the narrow portion 106, the diameter D₁ of the narrowportion 106, and the transition lengths L₂, L₃. All dimensions are shownin inches. Table 1 shows the result of a steel rod having fixedtransition lengths L₂, L₃ of 1.5 inch. The diameter D₁ of the narrowportion 106 ranged from 0.25 inch to 0.4375 inch. The length L₁ of thenarrow portion 106 was always started at 8 times the diameter D₁. Thepercent strain (or elongation of the yielding rod) was calculated asfollows:

% strain=(LE/L ₁)×100,   Eq. 2:

where LE is the length of elongation of the narrow portion 106 and L₁ isthe length of the narrow portion 106 prior to elongation.

TABLE 1 D₁ L₂, L₃ L₁ (8 × D₁₎ Elongation (LE) % strain 0.25 1.5 2 0.2 100.3125 1.5 2.5 0.23 9.2 0.375 1.5 3 0.29 9.666667 0.4375 1.5 3.5 0.339.428571

Table 2 shows the result of a steel rod having transition lengths L₂, L₃of 0.5 inch. The diameter D₁ of the narrow portion 106 ranged from 0.25inch to 0.4375 inch. The length L₁ of the narrow portion 106 was always8 times the diameter D₁.

TABLE 2 D₁ L₂, L₃ L₁ (8 × D₁) Elongation (LE) % strain 0.25 0.5 2 0.178.5 0.3125 0.5 2.5 0.22 8.8 0.375 0.5 3 0.26 8.666667 0.4375 0.5 3.50.31 8.857143

Table 3 shows the result of a steel rod having fixed transition lengthsL₂, L₃ of 0.5 inch. The diameter D₁ of the narrow portion 106 rangedfrom 0.25 inch to 0.4375 inch. The length L₁ of the narrow portion 106varied from 4 inches to 5.25 inches as shown in Table 3.

TABLE 3 D₁ L₂, L₃ L₁ Elongation (LE) % strain 0.25 0.5 4 0.35 8.750.3125 0.5 4.6875 0.42 8.96 0.375 0.5 5.25 0.46 8.761905 0.4375 0.5 5.250.47 8.952381

FIGS. 4A through 4C show the Mises stress (ksi) of one rod from thefinite element analysis. FIGS. 4A through 4C show the different amountsof stress on different portions of the rod 100 with increasing force inthe longitudinal direction. As shown in FIG. 4A, the stress is greatestat the narrow portion 106 and the junction 112. The amount of stressgradually decreases moving up the transition area 108 towards the firstportion 102 with the least amount of stress at the first portion.Similar results occurred in all rods 100 tested.

Based on the extensive experimentation, it was determined that in thepreferred embodiment, the length L₁ of the narrowed portion 106 shouldbe approximately eight to approximately sixteen times the diameter D₁ ofthe narrowed portion 106. However, the length L₁ did not have as mucheffect on the maximum strains as did the lengths L₂, L₃ of thetransition areas 108, 110. For example, a rod 100 with a reduceddiameter of 0.25 inch and a length of 2 inches (8 times its diameter)achieved an elongation of 0.17 inch, which translated into an 8.5%strain. A rod 100 with a reduced diameter D₁ of 0.25 inches and a lengthL₁ of 4 inches (16 times its diameter) achieved an elongation of 0.35inches, which translated into an 8.75% strain. Therefore, the length L₁of narrow portion 106 should have a minimum length of 8 times itsdiameter D₁ and can be adjusted to greater lengths based on the desiredelongation for the connection.

Surprisingly, it was the transition lengths L₂, L₃ that appeared to havea significant impact on the maximum strain on the yielding rod 100. Eachtapering portion or transition area occurs over a transition length L₂,L₃ of approximately one-quarter (0.25) inch to approximately two (2)inches. After extensive experimentation, it was determined that thetransition lengths L₂, L₃ are half (0.5) inch to approximately one and ahalf (1.5) inch were preferable. Most preferably, transition lengths L₂,L₃ are approximately three-quarters (0.75) inch.

The transition areas 108, 110 ensure that the axial stresses aresmoothly transferred to the narrowed portion L₁ and avoid prematurefracture due to concentrated stresses at the junctions 112, 114 wherethe transition areas 108, 110 meet the narrow portion 106. The junctions112, 114 also serve as a point at which the onset of yielding begins.After these junctions 112, 114 yield, it will experiencestrain-hardening and allow the entirety of the narrowed portion 106 toyield and exhibit plastic behavior. The junctions 112, 114 may be sharpor abrupt transitions from the respective transitional areas 108, 110 tothe narrowed portion 106 or they may be smooth, curved transitions fromthe transitional areas 108, 110 to the narrowed portion 106.

In order to determine the range of angles for the transitional areas108, 110 relative to the longitudinal axis A, a first analysis wasconducted using a 1.5 inch for each transition length L₂, L₃ anddifferent reduced diameters D₁ with a length L₁ of the narrowed portionat 8 times its diameter D₁. This established the lower bounds of theranges of angles.

It was determined from the analysis that the maximum angle for the slopeS (defined as the angle between the surface of the transitional area andthe longitudinal axis) of the transitional area 108, 110 should notexceed approximately 30 degrees. It was determined that any anglegreater than approximately 30 degrees will begin to significantly reducethe ductility of the invention due to increased concentrated stresses atthe transition area by increasing the stress concentrations at thejunction 112, 114 where the transition areas 108, 110 meet the narrowportion 106. Therefore, in the preferred embodiment, the slope of thetransition area relative to the longitudinal axis is preferablyapproximately 30 degrees or less. However, having transition area, 108,110 is better than not having transition areas; therefore, the yieldingrod 100 can have a transition area 108, 110 with a slope S of less than90 degrees, which will still work better than a rod without a transitionarea.

In the preferred embodiment, the first portion 102 and the secondportion 104 may be partially or completely threaded. The threadingallows the rod to receive a coupler 120 for securement and attachmentpurposes. Threading also allows the yielding rod 100 to be quickly andeasily replaced or repaired. For example, if during an earthquake theyielding rod 100 brakes at the narrowed portion 106 (since it is theweakest portion of the rod and designed to be weaker than the breakoutor pullout strength of the concrete) the yielding rod 100 can berepaired by cutting out the damaged or broken narrowed portion 106 andthe transition areas 108, 112 and replacing it with a new truncatedyielding rods 101 as shown in FIG. 3. The new yielding rod 101 alsohaving threaded ends can be secured to the old yielding rod 100 stillembedded in the foundation at one portion 104 with a coupler 120 andsecured to the yielding rod 100 still embedded in the building 10 at theopposite portion 102 with a second coupler 121.

The yielding rod 100 is preferably made out of material that exhibitductile behavior as defined by the American Concrete Institute®. Thepreferred material should have a tensile test elongation of at leastapproximately 14 percent and reduction area of at least approximately 30percent. Currently for threaded rods the following materials qualify asductile steels: ASTM A307, ASTM F1554, ASTM A193; and for deformedreinforcement: ASTM A615, ASTM A706 and ASTM A955. It is also desirablethat the material used have a narrow range of tensile strengths. Forexample, ASTM F1554, Gr. 36, material has an allowable tensile strengthrange of 58,000 psi to 80,000 psi. For optimal design and predictablebehavior, the range should be kept within 20%. For example, if ASTMF1554, Gr. 36, material is used, the manufacturer should ensure that theupper limit of the tensile strength should be 69,600 psi, instead of80,000 psi.

To manufacture a proper yielding rod for anchors used in existingfoundations, based on the existing foundation information, the maximumbreakout strength of the concrete and the maximum pullout strength ofthe anchor in tension must be calculated.

The lesser of the breakout strength of the concrete and the pulloutstrength of the anchor in tension is used to calculate the requireddiameter D₁ of the narrowed portion 106. In calculating the diameter D₁of the narrowed portion 106, the maximum probable strength of thematerial is used. For example, if the material has a specified tensilestrength ranging between 58,000 psi and 70,000 psi, use 70,000 psi. Thiswill ensure that the strength of the steel will still govern if thesteel provided has tensile strength at the upper limit of its allowablerange.

Based on the reduced diameter calculated above, calculate the designstrength of a single anchor or group of anchors by multiplying thenominal strength of a single anchor or group of anchors in tension, Nsa,with the strength reduction factor for ductile steel elements intension, F, as governed by the steel strength. F×Nsa should becalculated using the lower value of the specified tensile strength ofthe material. For example, if the material has a specified tensilestrength ranging between 58,000 psi and 70,000 psi, use 58,000 psi. Thisvalue will be the limiting strength of the connection and is to be usedin checking against the factored tensile force applied to an anchor orgroup of anchors, Nua, determined from analysis of the structure.

The length L₂, L₃ of the transitions areas 108, 110 and the narrowportion 106 based on physical constraints and desired performance forits intended use can be determined For example, if there is not muchspace, use a shorter length L₁ for the narrow portion 106. If there isplenty of space available and it is desirable to get more elongationcapacity out of the anchor, a longer length L₁ for the narrow portion106 can be used.

For anchors used in new foundations, the required diameter of thenarrowed portion 106 is calculated based on the factored tensile force,Nua, determined from analysis of the structure.

By way of example only, to use the yielding rod in an existingconstruction, a hole is drilled into the existing foundation, the holehaving a diameter that is prescribed by the epoxy manufacturer's report.The hole as is cleaned as prescribed by the epoxy manufacturer's report.The hole is filled with epoxy adhesive that complies with building coderegulations. One end 104 of the yielding rod 100 is inserted into theepoxy, such that the one transition area 110, the narrow portion 106,and a portion of one end 104 are all projecting out of the concrete 20.The other end 102 can be secured to an anchorage device 30 whichattaches to the structure 10.

In situations where a yielding rod 100 is being repaired, or a genericthread rod is being replaced, a threaded portion of the rod 104projecting out from the concrete foundation 210 can be cut and filed ifnecessary. One end 104 of the new yielding rod 101 can be connected tothe existing threaded rod 100 with a coupler 120. The coupler 120 may bebolt having internal threads to screw onto the threaded rod 100 at oneend and the yielding rod 101 at the other end. This process can berepeated at the opposite end of the yielding rod 101 connected to athreaded rod 102 projecting from the building 10 or house. In someembodiments, the coupler 120 may have two different diameters at theopposite ends to accommodate an existing threaded rod 100 and a yieldingrod 101 that may have two different diameters D₂.

By way of example only, to use the yielding rod in a new construction,place the rod 100 with a template at the location where the bolt willanchor down the structure, such that the one end 104 will be embedded inthe concrete 20. Provide a nut and plate washer at the bottom of thefirst end. Ensure that the rod 100 is placed as vertical as possible andensure that the other end 102, the transitional areas 108, 110, and thenarrowed portion 106 will all project outside of the concrete. Pour theconcrete and secure the yielding rod 100 to the building 10.

The foregoing description of the preferred embodiment of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed. Many modifications and variations are possiblein light of the above teaching. It is intended that the scope of theinvention not be limited by this detailed description, but by the claimsand the equivalents to the claims appended hereto.

1. A yielding rod defining a longitudinal axis, comprising: a. a firstportion terminating at a first end having a first cross-sectionaldiameter; b. a second portion opposite the first portion, the secondportion terminating at a second end and having a second cross-sectionaldiameter; c. a narrow portion in between the first and second portions,the narrow portion have a third cross-sectional diameter and a length;d. a first transition area in between and connecting the first portionand the narrow portion; and e. a second transition area in between andconnecting the second portion and the narrow portion, f. wherein thethird cross-sectional diameter is smaller than the first and secondcross-sectional diameters, g. wherein the first transition area and thesecond transition area each have a taper that gradually narrows towardsthe narrow portion, and h. wherein the length of the narrow portion isgreater than the third cross-sectional diameter, i. wherein the firstand second transition areas each have a transition length ofapproximately 0.25 inch to approximately 2 inches, j. wherein the firstand second transition areas each have a slope relative to thelongitudinal axis greater than zero degrees and less than approximately90 degrees, k. wherein the length of the narrow portion is at leastapproximately 8 times the diameter of the third cross-sectionaldiameter, and l. wherein the narrow portion has a static strength intension that is less than a static concrete breakout strength in tensionand a static pullout strength in tension of a concrete foundation inwhich the yielding rod is embedded.
 2. The rod of claim 1, wherein thefirst and second transition areas each have a transition length ofapproximately 0.5 inch to approximately 1.5 inch.
 3. The rod of claim 1,wherein the first and second transition areas have a slope relative tothe longitudinal axis of approximately 30 degrees or less.
 4. The rod ofclaim 2, wherein the length of the narrowed portion is between about 8times and about 16 times the diameter of the third cross-sectionaldiameter.
 5. A rod defining a longitudinal axis, comprising: a. a firstportion terminating at a first end having a first cross-sectionaldiameter; b. a second portion opposite the first portion, the secondportion terminating at a second end and having a second cross-sectionaldiameter; c. a narrow portion in between the first and second portions,the narrow portion have a third cross-sectional diameter and a length;d. a transition area in between and connecting the first portion and thenarrow portion, e. wherein the third cross-sectional diameter is smallerthan the first and second cross-sectional diameters, f. wherein thetransition area has a taper that gradually narrows towards the narrowportion, and g. wherein the length of the narrow portion is greater thanthe third cross-sectional diameter.
 6. The rod of claim 5, wherein thetransition area has a transition length of approximately 0.25 inch toapproximately 2 inches,
 7. The rod of claim 5, wherein the transitionarea has a transition length of approximately 0.5 inch to approximately1.5 inches.
 8. The rod of claim 5, wherein the transition area has aslope relative to the longitudinal axis greater than zero degrees andless than 90 degrees.
 9. The rod of claim 5, wherein the first andsecond transition areas have a slope relative to the longitudinal axisof approximately 30 degrees or less.
 10. The rod of claim 5, wherein thelength of the narrow portion is at least approximately 8 times thediameter of the third cross-sectional diameter.
 11. The rod of claim 5,wherein the length of the narrow portion is between about 8 times andabout 16 times the diameter of the third cross-sectional diameter. 12.The rod of claim 5, wherein the narrow portion has a static strength intension that is less than a static concrete breakout strength in tensionand a static pullout strength in tension of a concrete foundation inwhich the yielding rod is embedded.
 13. A rod defining a longitudinalaxis, comprising: a. a first portion having a first end and a secondend, and a first cross-sectional diameter; b. a narrow portion having asecond cross-sectional diameter and a length; c. a transition areaoperatively connecting the first portion to the second portion, thetransition area having a taper, d. wherein the second cross-sectionaldiameter is smaller than the first cross-sectional diameters, and e.wherein the taper of the transition area gradually narrows towards thenarrow portion forming a slope of approximately 30 degrees or lessrelative to the longitudinal axis.
 14. The rod of claim 13, wherein thetransition area has a transition length of approximately 0.25 inch toapproximately 2 inches,
 15. The rod of claim 13, wherein the transitionarea has a transition length of approximately 0.5 inch to approximately1.5 inches.
 16. The rod of claim 13, wherein the length of the narrowportion is at least approximately 8 times the diameter of the secondcross-sectional diameter.
 17. The rod of claim 13, wherein the length ofthe narrow portion is between about 8 times and about 16 times thediameter of the second cross-sectional diameter.
 18. The rod of claim13, wherein the narrow portion has a static strength in tension that isless than a static concrete breakout strength in tension and a staticpullout strength in tension of a concrete foundation in which theyielding rod is embedded.